Theorem pwex 3973

Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Hypothesis

Ref	Expression
zfpowcl.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
pwex	⊢ 𝒫 𝐴 ∈ V

Proof of Theorem pwex

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfpowcl.1	. 2 ⊢ 𝐴 ∈ V
2		pweq 3403	. . 3 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
3	2	eleq1d 2151	. 2 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∈ V ↔ 𝒫 𝐴 ∈ V))
4		df-pw 3402	. . 3 ⊢ 𝒫 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑧}
5		axpow2 3970	. . . . . 6 ⊢ ∃𝑥∀𝑦(𝑦 ⊆ 𝑧 → 𝑦 ∈ 𝑥)
6	5	bm1.3ii 3919	. . . . 5 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧)
7		abeq2 2191	. . . . . 6 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧))
8	7	exbii 1537	. . . . 5 ⊢ (∃𝑥 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧))
9	6, 8	mpbir 144	. . . 4 ⊢ ∃𝑥 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧}
10	9	issetri 2617	. . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑧} ∈ V
11	4, 10	eqeltri 2155	. 2 ⊢ 𝒫 𝑧 ∈ V
12	1, 3, 11	vtocl 2662	1 ⊢ 𝒫 𝐴 ∈ V

Syntax hints:   ↔ wb 103  ∀wal 1283   = wceq 1285  ∃wex 1422   ∈ wcel 1434  {cab 2069  Vcvv 2610   ⊆ wss 2982  𝒫 cpw 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612  df-in 2988  df-ss 2995  df-pw 3402

This theorem is referenced by:  pwexg  3974  p0ex  3979  pp0ex  3980  ord3ex  3981  abexssex  5803  npex  6777  axcnex  7141  pnfxr  7285  mnfxr  7289  ixxex  9050